Strategic Inattention in the Sir Philip Sidney Game

bioRxiv preprint doi: https://doi.org/10.1101/559955; this version posted February 25, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

STRATEGIC INATTENTION IN THE SIR PHILIP SIDNEY GAME

By Mark Whitmeyer1 Infamously, the presence of honest communication in a signaling envi- ronment may be diﬃcult to reconcile with small signaling costs or a low degree of common interest between sender and receiver. We argue that one mechanism through which such communication can arise is through inattention on the part of the receiver, which allows for honest communication in settings where, should the receiver be fully attentive, honest communication would be impossible. We explore this idea through the Sir Philip Sidney game in detail and show that some degree of inattention is always weakly better for the receiver, and may be strictly better. Moreover, some inattention may be a Pareto improvement and leave the sender no worse oﬀ. We compare limited attention to Lachmann and Bergstrom’s (1998) notion of a signaling medium and show that the receiver-optimal degree of inattention is equivalent to the receiver-optimal choice of medium.

Keywords: Costly Signalling, Strategic Inattention, Handicap Theory, Information Design

1. Introduction

I have only one eye–I have a right to be blind sometimes. ...I really do not see the signal.

Admiral Horatio Lord Nelson The handicap principle is an important notion in signalling games. Put simply, this principle states that order to facilitate meaningful communication in situations in which there are conﬂicts of interest, a cost is necessary. One standard setting for formally investigating this principle is the discrete Sir Philip Sidney Game (Maynard Smith [13]). In the classic formulation of this game, two players interact, a sender and a receiver. The sender’s type is uncertain: he is either healthy (with probability 1 − µ) or needy (with probability µ). The sender is the ﬁrst mover, and may choose to either cry out and incur a cost of c > 0 or stay silent and incur no cost. Following this action (henceforth referred to as a signal) by the sender, the receiver observes the signal and then chooses whether to donate a resource and incur a cost of d > 0 or to do nothing and incur no cost. Should a sender receive a donation, his probability of survival is 1 regardless of his type. On the other hand, if a sender does not receive a donation then his probability of survival is 1 − a if he is needy and 1Department of Economics, University of Texas at Austin Email: [email protected]. Thanks to Rosemary Hopcroft, Ryan Oprea, Joseph Whitmeyer, and Thomas Wise- man. All remaining errors are my own. 1 date: February 24, 2019 bioRxiv preprint doi: https://doi.org/10.1101/559955; this version posted February 25, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

1 − b if he is healthy, where a > b. In addition, there is a relatedness parameter k ∈ [0, 1] that captures the degree of common interest between the sender and the receiver–under any vector of strategies each player receives k times the payoff of the other player plus his or her own payoff. The purpose of this paper is to explore strategic inattention in this setting. That is, in recent work, Whitmeyer (2019) [15] notes that full transparency is not generally optimal for the receiver in signalling games, and takes steps to characterize the receiver-optimal degree of transparency in those games. In particular, less than full transparency may beget separating equilibria (equilibria in which different types send different signals) in situations where there is little to no meaningful communication under full transparency.1 In the Sir Philip Sidney game, such a breakdown in communication occurs for certain regions of the parameters corresponding to a low cost of crying out, a low degree of relatedness, and a low cost of donation d. However, not all is lost. We embed the game into a slightly larger game, one with an additional (first stage) in which the receiver chooses a level of attentiveness in the signalling game. She chooses a probability x, such that in the signalling portion of the game she observes the sender’s choice of signal with probability x and does not with probability 1 − x, which probabilities are independent of the signal choice of the sender. Remarkably, some degree of inattention is always (weakly) optimal for the receiver: in certain regions of the parameter space the receiver is strictly better off when she is partially inattentive, and it may also be a Pareto improvement, leaving the sender himself no worse off. Inattention is helpful in the following two ways. First, it provides a lower bound on the set of equilibrium payoffs of the game: since complete inattention may be chosen in the initial stage, at equilibrium the receiver can do no worse than the unique equilibrium payoff in the signalling game given complete inattention, where the two types of sender both remain silent. Second, and perhaps more compellingly, there is an interval of the attention parameter in which a separating equilibria may manifest, despite the non-existence of such an equilibrium in the game with full attention. Put simply, inattentiveness enhances communication.

1.1. Related Work

There is a substantial literature in theoretical biology, philosophy, and economics exploring signalling games, commencing with Lewis (1969) [12]. In the same era, Zahavi (1975) [18] published his seminal work on handicaps in biology, and this notion was later incorporated in signalling games themselves. The discrete Sir Philip Sidney Game (Maynard Smith (1991) [13]) has become one of the paradigmatic settings for investigating the handicap principal in signalling games. Numerous other papers investing this and other closely related games have ensued, and if ever an area could be termed burgeoning it is this one. A list of recent works includes Bergstrom and Lachmann (1998) [2], Huttegger and Zollman

1This has been noted by Lachmann and Bergstrom (1998) [10] in the context of the continuous Sir Philip Sidney game.

date: February 24, 2019 bioRxiv preprint doi: https://doi.org/10.1101/559955; this version posted February 25, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

(2010) [6], Huttegger, Skyrms, Tarrès, and Wagner (2014) [5]. Also pertinent is the survey article on the handicap principle, Számadó(2011) [14]. In recent work, Zollman, Bergstrom and Huttegger (2013) [19] highlight the fact that empirically, “re- searchers have not always been able to find substantial signal costs associated with putative costly signal systems–despite evidence that these systems do convey, at least, some information among individuals with conflicting interests”, and ask, “What then, are we to make of empirical situations in which signals appear to be informative even without the high costs required by costly signalling models?” As they mention, other works have illustrated–see e.g. Lachmann, Számadó,and Bergstrom (2001)[11]–that this issue may be ameliorated by recognizing that costly signals need not be sent on the equilibrium path, and that it is the high cost of sending a (deviating) signal that keeps the senders honest. In addition, they forward an alternate resolution of the issue: there are also partially informative equilibria (in which players mix), which may be sustained despite low or non-existent costs. Hence, in the same vein, we explore a third possibility: that limited attention may explain the existence of honest signalling, even with low costs. The paper closest in spirit to this one is Lachmann and Bergstrom (1998) [10]. There, the authors allow for perceptual error on the part of the receiver and introduce the notion of a medium, which distorts the signals observed by the receiver. They illustrate that different media may beget different equilibria, and that some media may even foster honest communication impossible in other media. In this paper we endogenize the medium by making it a choice of the receiver. Moreover, we restrict the set of media the receiver can choose to those of a specific sort: those which correspond to inattention. However, as we show in Theorem 3.11, this is no restriction, the receiver-optimal equilibrium under her optimal choice of attention remains supreme even were she able to choose any medium, however complex. Other papers that allow for perceptual error, or noise, include Johnstone and Grafen (1992) [9], Johnstone (1994, 1998) [7,8], Lachmann, Számadó,and Bergstrom (2001)[11], and Wiley (2013, 2017) [16, 17].

2. The Classic Sir Philip Sidney Game

We begin by revisiting the (more-or-less standard) Sir Philip Sidney game (where we allow the cost, c = 0), which game is depicted in Figure 1. There are two players, a sender (he) and a receiver (she);

and the sender is one of two types, healthy or needy: Θ = {θH , θN }. The sender’s type is his private

information, about which both sender and receiver share a common prior, µ := Pr (Θ = θN ). After being informed of his type, the sender chooses to either cry out (cry) or stay quiet (quiet). The receiver observes the sender’s choice of signal (but not his type), updates her belief about the sender’s type based on her prior belief and the equilibrium strategies and elects to either donate a resource (donate) or refuse to donate (decline). We impose that a > b and that a, b, c, and d take values in the

(1 − c, 1 − d) (1, 1 − d) donate donate cry θH quiet

decline decline (1 − b − c, 1) 1 − µ (1 − b, 1)

R R

µ (1 − c, 1 − d) (1, 1 − d) donate donate cry quiet

θN (1 − a − c, 1) decline decline (1 − a, 1)

Fig 1. The Sir Philip Sidney Game

interval [0, 1]. There is also a relatedness parameter k ∈ [0, 1]: after each outcome, a player receives his own payoff plus k times the payoff of the other player. Throughout, we impose the following conditions: a > d/k > b and a > b > dk + c. The first assumption ensures that if the receiver is (sufficiently) confident that the sender is healthy then she strictly prefers not to donate and if she is sufficiently confident that the sender is needy then she strictly prefers to donate. The second assumption eliminates any separating equilibria. In addition, we define dˆ:= k (µa + (1 − µ) b), which will thankfully save some room on the manuscript. We describe the equilibrium in the signalling game as a four-tuple (·, ·; ·, ·), where the first entry corre-

sponds to the strategy of θH , the second entry to the strategy of θN , the third entry to the response of the receiver to quiet, and the fourth entry to the response of the receiver to cry. In the case of pooling equilibria (equilibria in which both types of sender choose the same signal), we leave the response of the receiver to an oﬀ-path signal as · when there may be multiple responses that would sustain an equilibrium. We have the following result, which follows from our parametric assumptions:

Lemma 2.1. There exist no separating equilibria.

Proof. Standard see e.g. Bergstrom and Lachmann (1997) [1].

There do; however, exist pooling equilibria, both those in which both senders choose cry and those in which both senders choose quiet. Note that the pooling equilibrium in which both senders choose cry requires that the receiver’s belief upon observing quiet (an oﬀ-path action) be such that she would at least (weakly) prefer to choose decline rather than donate. Moreover, the pooling equilibrium in which both senders choose quiet and to which the receiver responds with decline also requires that the receiver’s belief upon observing cry (an oﬀ-path action) be such that she would at least (weakly) prefer to choose decline rather than donate. In some sense, this is less convincing of an equilibrium: shouldn’t

the needy bird be more likely to cry out?2 The other pooling equilibrium, that in which both senders choose quiet and the receiver responds with donate makes no restrictions on the receiver’s oﬀ-path beliefs and is in that sense quite strong. Formally,

Lemma 2.2. There exist pooling equilibria: (cry, cry; ·, decline) is never an equilibrium. If d ≤ dˆ then (cry, cry; ·, donate) is an equilibrium, given the appropriate off-path beliefs for the receiver; and (quiet, quiet; donate, ·) is an equilibrium, regardless of the receiver’s off-path beliefs. If d ≥ dˆ, then (quiet, quiet; decline, ·) is an equilibrium, given the appropriate off-path beliefs for the receiver.

Proof. Standard, see e.g. Bergstrom and Lachmann (1997) [1].

There also exist equilibria in which players mix. We omit those equilibria since the receiver’s payoﬀ is higher under the best pure-strategy (pooling) equilibrium than under one in which at least one sender mixes.3 If d ≥ dˆ then the receiver optimal equilibrium is (quiet, quiet; decline, ·), which yields her a payoﬀ of

VR = 1 + k (1 − b) − kµ(a − b) (1)

and if d ≤ dˆ then the receiver optimal equilibrium is (quiet, quiet; donate), which yields her a payoﬀ of

VR = 1 − d + k (2)

3. Strategic Inattention

We now explore the notion that full attention may not be generally optimal for the receiver. We modify the game by introducing an initial stage in which the receiver chooses and commits to a level of attention. This is modeled in the following parsimonious fashion: in the ﬁrst stage the receiver chooses an Attention Parameter x, which is publicly observable,4 and then the signalling game proceeds in the standard manner. See Figure 2. The attention parameter is straightforward: the receiver simply observes the sender’s choice of signal with probability x and does not observe the choice with probability 1 − x. If the receiver does not observe the signal then she simply makes the optimal choice given her information, which is merely her prior. Hence, she chooses decline if and only if d ≥ dˆ and donate otherwise. The game has two stages: 1) the receiver chooses x followed by 2) the signalling game under parameter x. Hence, we search for subgame perfect equilibria through backward induction.

2Cf. the “Intuitive Criterion” as developed in Cho and Kreps (1987) [3]. 3This follows from the proof of Theorem 3.5, infra, contained in Appendix A.2. 4Alternatively, we can think of x as being an exogenous primitive of the model, and thus our endeavor may be reinterpreted as searching for the optimal level of attention for the receiver.

nature

θH θN R

x x

cry quiet cry quiet

......

Fig 2. Choosing an Attention Parameter

3.1. The Signalling Subgame

Let us examine the subgame that consists of the signalling game for a given choice of attention parameter x. As we discover, there are two critical cutoﬀ beliefs of x, x andx ¯, which divide the range of possible ¯ values of x into three regions, depicted in Figure3. Explicitly,

c c x := , x¯ := ¯ a − dk b − dk A := [0, x), B := [x, x¯], C := (¯x, 1] ¯ ¯ Our ﬁrst result highlights that, in contrast to the ﬁrst section of this paper, in which there did not exist separating equilibria, other values of x may beget separation. Viz,

Lemma 3.1. If x ∈ B then there exists a separating equilibrium in which θH chooses quiet and θN chooses cry. The level of attention that maximizes the receiver’s payoﬀ is x∗ = c/(b − dk) =x ¯. These equilibria yield payoﬀs to the receiver of

VR(x) = 1 + k (1 − b) − kµ (a + c − b)+ µx (ak − d) (3)

for d ≥ dˆ, and

VR(x) = 1 − d + k (1 − cµ) + (1 − µ)x (d − bk) (4)

for d ≤ dˆ.

A B C

x x¯ 0 ¯ 1

Fig3 . The Critical Regions of x

Proof. Proof is left to Appendix A.1.

The intuition behind this result is simple. By being strategically inattentive (choosing x within B), the parent can lessen the incentive of either type to deviate and mimic the other. The receiver-optimal ∗ attention parameter x is that which leaves type θH indiﬀerent between separating and deviating to

mimic θN . The other separating equilibrium remains unattainable, however. Indeed,

Lemma 3.2. There exists no attention parameter x ∈ [0, 1] such that there exists a separating

equilibrium in which θH chooses cry and θN chooses quiet.

Proof. Suppose the two types of sender separate and let θH choose cry and θN choose quiet. ˆ Suppose ﬁrst that d ≥ d, so that decline is the receiver’s response to not observing. But then type θH ’s incentive constraint is

1 − b − c + k ≥ x (1 + (1 − d) k) + (1 − x) (1 − b + k)

Or, (dk − b) x ≥ c, which is impossible due to our assumed conditions. Next, suppose d ≤ dˆ, so that

decline is the receiver’s response to not observing. But then type θH ’s incentive constraint is

x (1 − b − c + k) + (1 − x) (1 − c + (1 − d)k) ≥ (1 + (1 − d) k)

Or, (dk − b) x ≥ c. Again, impossible.

Of course, there exist pooling equilibria as well. Viz,

Lemma 3.3. There does not an exist an x ∈ [0, 1] such that (cry, cry; ·, decline) is an equilibrium. Let d ≥ dˆ. (quiet, quiet; decline, ·) is an equilibrium regardless of the receiver’s off-path beliefs provided x ≤ c/(a − dk), and is an equilibrium provided the appropriate off-path beliefs for the receiver otherwise. The receiver’s resulting payoff is given in Expression 1. If d ≤ dˆ and x ≥ c/(b − dk) then (cry, cry; ·, donate) is an equilibrium provided the appropriate off-path beliefs for the receiver. For d ≤ dˆ, (quiet, quiet; donate, ·) is also an equilibrium, regardless of the receiver’s off-path beliefs. The receiver’s resulting payoff is given in Expression 2.

Proof. Let d ≥ dˆ. Then, the receiver’s optimal action should she choose not to observe a signal is decline. From the analysis in Lemma 2.2, we may immediately conclude that regardless of x, there is no equilibrium in which the two types of sender pool on cry.

Next, we examine (quiet, quiet; decline, ·). It is clear that the oﬀ-path belief that leaves the equilibrium in greatest jeopardy is that which insists the receiver prefer donate upon observing cry. Under

this “worst” case scenario, the incentive constraint for θH is

1 − b + k ≥ x (1 − c + (1 − d)k) + (1 − x) (1 − b − c + k)

Or, c/(b − dk) ≥ x. Analogously, the incentive constraint for θN reduces to c/(a − dk) ≥ x. Thus, if x ≤ c/(a − dk) then regardless of the receiver’s oﬀ-path beliefs, (quiet, quiet; decline, ·) is an equilibrium. If x is above this threshold, then it is clear that an oﬀ-path belief that results in the receiver (weakly) preferring decline upon observing cry is required. Now, let d ≤ dˆ. The receiver’s optimal action should she choose not to observe a signal is donate.

First, we explore whether there is an equilibrium in which the two types of sender pool on cry. For θH we have 1 − c + (1 − d)k ≥ x (1 − b + k) + (1 − x) (1 + (1 − d) k)

which holds provided x ≥ c/(b − dk). For θN we have

1 − c + (1 − d)k ≥ x (1 − a + k) + (1 − x) (1 + (1 − d) k)

which holds provided x ≥ c/(a − dk). Note that here we have assigned the receiver’s oﬀ-path belief to be such that decline is a (weak) best response to quiet. This is clearly necessary for the existence of this equilibrium, irrespective of x. Finally, suppose the two types of sender pool on quiet. Again, it is clear that this is an equilibrium, regardless of x or the oﬀ-path beliefs.

As this proof illustrates, if x is suﬃciently low, there is no equilibrium that consists of (cry, cry; ·, donate). Indeed,

Corollary 3.4. Let x < c/(b − dk). Then there exist no pooling equilibria in which each type of sender chooses cry.

Provided x∈ / A, there also exist equilibria in which at least one player mixes. However, as in the previous section, those equilibria do not maximize the receiver’s payoﬀ.5

Theorem 3.5. If x ∈ A, then the unique equilibrium is (quiet, quiet; decline, ·), where · is the optimal response of the receiver to any belief given an oﬀ-path action of cry; and if x ∈ B, then (quiet, quiet; decline, ·) and (quiet, cry; decline, donate) are the two possible pure strategy equilibria. If x ∈ C, then (quiet, quiet; decline, ·) is the unique pure strategy equilibrium (given appropriate oﬀ-path beliefs). 5This follows from the proof of Theorem 3.5, contained in Appendix A.2.

Proof. For completeness, this proof may be found in Appendix A.2. The proof is straightforward, though tedious, and consists simply of the enumeration of all of the possible equilibria and the conditions

under which they exist.

3.2. The First Stage

Armed with our analysis from the preceding subsection, we may now characterize the subgame perfect equilibria of the game.

Lemma 3.6. Let d ≥ dˆ and x ∈ B. Then the equilibrium that maximizes the receiver’s payoff is (quiet, cry; decline, donate) if and only if x ≥ kc/(ak − d). Otherwise it is (quiet, quiet; decline, ·). Let d ≤ dând x ∈ B. Then the equilibrium that maximizes the receiver’s payoff is (quiet, cry; decline, donate) if and only if x ≥ (kcµ)/ ((1 − µ) (d − bk)). Otherwise it is (quiet, quiet; donate, ·).

Proof. First, let d ≥ dˆ. Using the receiver’s payoﬀ from the pooling equilibrium (Expression 1) and her payoﬀ from the separating equilibrium (Expression 3), we have

sep pool VR (x) ≥ VR (5) 1 + k (1 − b) − kµ (a + c − b)+ µx (ak − d) ≥ 1 + k (1 − b) − kµ (a − b)

Or,

kc x ≥ (6) ak − d

Note also that if x = x∗ = c/(b − dk) then this simpliﬁes to (ak − d) ≥ k (b − dk). Second, let d ≤ dˆ. Then,

sep pool VR (x) ≥ VR (7) 1 − d + k (1 − cµ)+ x(1 − µ) (d − bk) ≥ 1 − d + k

Or, kcµ x ≥ (8) (1 − µ)(d − bk) ∗ If x = x then this simpliﬁes to (1 − µ) (d − bk) ≥ µk (b − dk).

Pausing briefly to look at the cutoffs above which separation is better, we see that the right-hand side of Inequality 6 is increasing in both c and d and decreasing in k and a. Hence, small6 decreases in the signalling and/or donation costs enlarge the set of attention parameters x such that separation is better for the receiver than pooling. Analogously, small increases in the relatedness parameter and/or the cost suffered by the needy type have the same effect.

6The modiﬁer “small” is required since we have already imposed several conditions on the values that the parameters may take.

The right-hand side of Inequality 8 is also increasing in c and decreasing in k, and so the same intuition holds. However, it is now increasing in b and in µ and decreasing in d. It is easy to see why it should be increasing in µ: as the proportion of needy types increases, the uninformativeness of pooling is not as harmful to the receiver (recall that since d ≤ dˆ the receiver is donating). Similar reasoning explains the relationship with d: as d increases, pooling becomes more costly since donation itself is more costly. As b increases it becomes harder to elicit separation (from the high type), which reduces the receiver’s beneﬁt from the separating equilibrium. Note that the right hand side of Inequality 6 is greater than the right hand side of Inequality 8 if and only if d ≥ dˆ. We introduce the following conditions

Condition 3.7. d ≥ dˆ and Inequality 6 holds.

Condition 3.8. d ≤ dˆ and Inequality 8 holds.

At long last, we characterize the subgame perfect equilibria of this game:

Theorem 3.9. Suppose there exists some xˆ ∈ B such that either Condition 6 or Condition 8 holds. Then, there exists a collection of subgame perfect equilibria consisting of a choice of x ≥ xˆ in the ﬁrst stage, and (quiet, cry, decline, donate) in the signalling portion of the game. The receiver optimal subgame perfect equilibria is that in which x = x∗ = c/(b − dk).

Proof. This follows from Lemma 3.6.

Theorem 3.10. There is always a collection of subgame perfect equilibria consisting of any choice of x in the ﬁrst stage, and (quiet, quiet, decline, ·) in the signalling portion. If neither Condition 3.7 nor Condition 3.8 holds then this collection is unique.

Proof. This follows from the fact that (quiet, quiet, decline, ·) is the equilibrium that (uniquely) maximizes the receiver’s payoﬀ. Since this is the unique equilibrium should the receiver choose any x ∈ A, this must be the equilibrium played for any x since otherwise the receiver would have a proﬁtable

deviation in the initial stage to an x ∈ A.

This pair of theorems evinces the two main effects of allowing the receiver to choose her level of attention initially. First, limited attention yields separating equilibria even when such equilibria could not exist under full attention. That is, honest communication is manifested in a scenario in which the conflict between the receiver and the sender would usually be too great. Second, enabling the receiver to choose her level of attention ensures that the equilibrium played in the signalling portion of the game is relatively “good” for the receiver (either best or second-best) and provides a lower bound on the receiver’s payoff.

Now, let us take a brief sojourn to look at the sender’s welfare. In particular, we look at his expected welfare from the ex ante perspective (that is, before he knows his type). Suppose ﬁrst that we are in the full attention setting, x = 1. Let d ≥ dˆ, and so for the equilibrium in which both types pool on quiet,

VS = (1 − µ) (1 − b + k) + µ (1 − a + k) = 1 − b + k − µ(a − b) (9)

This is; however, not the best equilibrium for the sender. Instead, the best equilibrium is (σH , cry; decline, λ),

where σH := Pr(cry|θH ) = µ(ak − d)/((1 − µ)(d − bk)), and λ := Pr(donate|cry) = c/(b − dk) are mixed strategies. This yields

VS = (1 − µ) (1 − b + k) + µ (λ (1 − c + (1 − d)k) + (1 − λ) (1 − a − c + k)) (10)

to the sender. Now, consider the separating equilibrium with x ∈ B. The sender’s ex ante payoﬀ is

VS = (1 − µ) (1 − b + k) + µ (x (1 − c + (1 − d)k) + (1 − x) (1 − a − c + k))

But this is the same expression as Expression 10! Moreover, x∗ = λ, and so we see that if the conditions for Theorem 3.9 hold, then attention parameter x = x∗ is actually a Pareto improvement over x = 1, since the receiver is made strictly better oﬀ and the sender is no worse oﬀ. However, it obvious that if d ≤ dˆ then the pooling equilibrium in which both types remain silent and are nevertheless donated to is optimal for the sender, and so no separating equilibrium engendered by attention x ∈ B could be a Pareto improvement.

3.3. Inattention Corresponds to the Receiver-Optimal Medium

Recall that in Bergstrom and Lachmann (1997) [10] the authors introduce the notion of a imperfect medium, which distorts signals. That is, the medium deﬁnes a conditional probability distribution of perceived signals dependent on which signals are actually sent. Here, we show that the optimal level of inattention, is equivalent to the best-possible medium for the receiver. To wit,

med Theorem 3.11. VR be the receiver’s payoﬀ for the receiver-optimal equilibrium under the best- med possible medium for the receiver. Then there exists an attention parameter x such that VR(x) = VR . If either Condition 3.7 or Condition 3.8 holds then the optimal parameter is x∗ = c/(b − dk). If neither holds then any parameter x ∈ [0, 1] is optimal.

Proof. We wish to choose a medium in order to maximize VR, which we will then show coincides with the receiver’s payoﬀ under inattention. From Whitmeyer (2019) [15] it is without loss of generality to restrict our attention to pure strategies of the senders. Moreover, as in [15], we consider a (relaxed) commitment problem for the receiver. That is, suppose that the receiver can commit to choosing donate with probability p and decline with probability 1 − p following cry; and donate with probability q and decline with probability 1 − q following quiet. The receiver solves the following optimization problem,

max {VR} p,q subject to

q (1 + (1 − d) k) + (1 − q) (1 − b + k) ≥ p (1 − c + (1 − d) k) + (1 − p) (1 − b − c + k)(IC1)

and

p (1 − c + (1 − d) k) + (1 − p) (1 − a − c + k) ≥ q (1 + (1 − d) k) + (1 − q) (1 − a + k)(IC2)

where

VR = (1 − µ) [q (1 − d + k) + (1 − q) (1 + (1 − b) k)]

+ µ [p (1 − d + (1 − c)k) + (1 − p) (1 + (1 − a − c) k)]

This optimization problem is easy to solve and yields q = 0 and p = c/(b − dk) for d ≥ dˆ, and ˆ ∗ q = 1 − c/(b − dk) and p = 1 for d ≤ d. Substituting these into the value function, we obtain VR (x ). If either Condition 3.7 or Condition 3.8 holds, then this maximizes the receiver’s payoff, and as illustrated in Whitmeyer (2019) [15], since p and q solve the relaxed (commitment) problem, and the payoff the receiver obtains under such p and q is attainable in the unrelaxed problem (which clearly holds since the receiver can obtain this payoff merely through limited attention), this must be the solution to the problem of choosing an optimal medium.7 On the other hand, if neither condition holds then the result is trivial. The receiver-optimal pooling equilibrium is attainable under any x and by the definition of the two conditions, such a pooling

equilibrium is best, should neither hold.

4. Discussion

The primary goal of this paper was to illustrate the counter-intuitive fact that limited attention may enhance honest communication in situations of conﬂict. The main results suggest several qualitative and hence testable implications. Broadly, these results imply that in situations in which there are severe conﬂicts of interest between the sender and receiver, we should expect inattentive receivers, even if there is no cost (of either an explicit or implicit sort) of attention. Moreover, such inattention may beget honest communication even in the absence of communication costs, or if such costs are surprisingly low. Furthermore, in such situations we may also expect to see pooling equilibria in which the senders choose the least cost signal, which signal the receiver may not even observe. The possibility of limited attention greatly strengthens the least cost pooling equilibria, since a receiver could always default to that by being very inattentive.

7See Whitmeyer (2019) for a more in depth exposition of this concept.

Another interpretation of this paper is to see it as highlighting the benefits of inattention to the receiver. That is, this work forwards an idea of the ilk, “if she were (involuntarily) inattentive, she would be better off”.8 This fits in well with the formulation of the initial stage in which the receiver chooses her attention level (which is observed by the sender). In other words, there is commitment to this level of attention that corresponds to the question, “if the receiver could choose her attention level in the signalling game, what would she choose?” It is important to note that without this commitment, the separating result would vanish and indeed the only equilibria that would remain would be the one in which both types of sender remain quiet. Of course, the separating equilibrium would reappear should the model be altered further so that attention is costly for the receiver, say increasing in x. Such a model is beyond the scope of this work, and perhaps merits further attention elsewhere. Finally, note that the solution concept used throughout this work is a refinement of the standard Nash Equilibrium, the Subgame Perfect Equilibrium, and not Evolutionary Stability. However, note that if x is fixed and in the interior of B, then there is a separating equilibrium that is strict, and thus must therefore be an Evolutionary Stable Strategy in the symmetrized game (see e.g. Cressman 2003 [4]). It might be interesting to explore the dynamic properties of the scenario from this manuscript in greater detail.

References

[1] Carl T. Bergstrom and Michael Lachmann. Signalling among relatives. i. is costly signalling too costly? Philosophical Transactions of the Royal Society B: Biological Sciences, 352(1353):609617, 1997. [2] Carl T. Bergstrom and Michael Lachmann. Signaling among relatives. iii. talk is cheap. Proceedings of the National Academy of Sciences, 95(9):5100–5105, 1998. [3] In-Koo Cho and David M. Kreps. Signaling games and stable equilibria. The Quarterly Journal of Economics, 102 (2):179–221, 1987. [4] Ross Cressman. Evolutionary Dynamics and Extensive Form Games. MIT Press, Cambridge, MA, 2003. [5] Simon Huttegger, Brian Skyrms, Pierre Tarr`es,and Elliott Wagner. Some dynamics of signaling games. Proceedings of the National Academy of Sciences, 111:10873–10880, 2014. [6] Simon M. Huttegger and Kevin J. S. Zollman. Dynamic stability and basins of attraction in the sir philip sidney game. Proceedings of the Royal Society B, 277, 2010. [7] Rufus A. Johnstone. Honest signalling, perceptual error and the evolution of all-or-nothing displays. Proceedings of the Royal Society B, 256, 1994. [8] Rufus A. Johnstone. Eﬃcacy and honesty in communication between relatives. The American Naturalist, 152(1): 45–58, 1998. [9] Rufus A. Johnstone and Alan Grafen. Error-prone signalling. Proceedings of the Royal Society B, 248, 1992. [10] Michael Lachmann and Carl T. Bergstrom. Signalling among relatives: ii. beyond the tower of babel. Theoretical Population Biology, 54(2):146 – 160, 1998.

8Note that, as is mentioned at the end of Section 3.2, for d ≥ dˆ, the best separating equilibrium for the receiver would leave the sender no worse oﬀ as well.

[11] Michael Lachmann, Szabolcs Számadó,and Carl T. Bergstrom. Cost and conflict in animal signals and human language. Proceedings of the National Academy of Sciences, 98(23):13189–13194, 2001. [12] David K. Lewis. Convention. A Philosophical Study. Harvard University Press, Harvard, 1969. [13] John Maynard Smith. Honest signalling: the philip sidney game. Animal Behavior, 42:1034–1035, 1991. [14] Szabolcs Szmad. The cost of honesty and the fallacy of the handicap principle. Animal Behaviour, 81(1):3 – 10, 2011. [15] Mark Whitmeyer. On optimal transparency in signaling. Mimeo, January 2019. [16] R. Haven Wiley. A receiver-signaler equilibrium in the evolution of communication in noise. Behaviour, 150(9/10): 9570993, 2013. [17] R. Haven Wiley. How noise determines the evolution of communication. Animal Behaviour, 124:307 – 313, 2017. [18] Amotz Zahavi. Mate selectiona selection for a handicap. Journal of Theoretical Biology, 53(1):205 – 214, 1975. [19] Kevin J. S. Zollman, Carl T. Bergstrom, and Simon M. Huttegger. Between cheap and costly signals: the evolution of partially honest communication. Proceedings of the Royal Society B, 280, 2013.

APPENDIX A: SECTION 2 PROOFS

Here, we present the proofs of selected results contained in the paper. Throughout, we denote mixed

strategies for each type of sender as σi := Pr(cry|θi) for i ∈ {H,N}, and for the receiver as λ := Pr(donate).

A.1. Lemma 1.3. Proof

Proof. Suppose the two types of sender separate: θH chooses quiet and θN chooses cry. First, let d ≥ dˆ, so that decline is the receiver’s response to not observing. Hence, the receiver’s payoﬀ is

VR = (1 − µ) (1 + (1 − b) k) + µ (x (1 − d + (1 − c) k) + (1 − x) (1 + (1 − a − c) k))

= 1 + k (1 − b) − kµ (a + c − b)+ µx (ak − d)

which is increasing in x. The senders’ incentive constraints are

1 − b + k ≥ x (1 − c + (1 − d) k) + (1 − x) (1 − b − c + k)

or c ≥ (b − dk) x for θH , and

x (1 − c + (1 − d) k) + (1 − x) (1 − a − c + k) ≥ 1 − a + k

or (a − dk) x ≥ c for θN . Thus, any x that satisﬁes

c c ≥ x ≥ (A1) b − dk a − dk begets a separating equilibrium of this form. Since a > b > dk + c, both a − dk and b − dk are greater than c. Hence, c x∗ = (A2) b − dk

and so µc (ak − d) V (x∗) = 1 + k (1 − b) − kµ (a + c − b)+ (A3) R b − dk When d ≤ dˆ, we simply proceed in the same manner and obtain the same interval and x∗ as in Inequality A1 and Equation A2, respectively. The resulting payoﬀ for the receiver is

(1 − µ)c (d − bk) V (x∗) = 1 − d + k (1 − cµ)+ (A4) R b − dk

A.2. Theorem 3.5. Proof

We proceed through a sequence of lemmata:

Lemma A.1. There exist no equilibria in which θH chooses a non-degenerate mixed strategy and θN chooses quiet.

ˆ Proof. First, let d ≥ d. Since θH is mixing, he must be indiﬀerent over his pure strategies in support. Hence, 1 − b − c + k = x (λ (1 + (1 − d) k) + (1 − λ) (1 − b + k)) + (1 − x) (1 − b + k)

where we have allowed the receiver to mix upon observing quiet (it is clear that following cry she will strictly prefer to choose decline). This reduces to xλ (dk − b)= c, which is impossible. ˆ Second, let d ≤ d. As above, for θH we have

x (1 − b − c + k) + (1 − x) (1 − c + (1 − d)k) = 1 + (1 − d)k

since it is clear that the receiver will prefer decline following cry and donate following quiet. This

reduces to x (dk − b)= c, which is always false.

Lemma A.2. Let d ≥ dˆ.

1. If x < c/(b − dk), there exists no equilibrium in which θH chooses a non-degenerate mixed strategy

and θN chooses cry.

2. If x = c/(b − dk), there exists a continuum of equilibria (σH , cry, decline, donate), where σH ≤

µ(ak − d)/((1 − µ)(d − bk)). The receiver’s payoﬀ is decreasing in σH and so is maximized at

σH = 0, which yields her a payoﬀ of VR = 1 + k (1 − b) − kµ (a + c − b)+ µx (ak − d).

3. If c/(b − dk) ≤ x ≤ 1, then there exists an equilibrium (σH , cry, decline, λ), where σH = µ(ak −

d)/((1 − µ)(d − bk)) and λ = c/ (x(b − dk)). The receiver’s payoﬀ is VR = 1 + k (1 − b) −

kµ (a + c − b)+ ckσH (1 − µ).

Proof. It is clear that following quiet the receiver will strictly prefer decline. Suppose that the receiver mixes and chooses donate with probability λ following cry (where λ ∈ [0, 1]), and hence captures

pure strategies for the receiver as well). Then, for θH to be indiﬀerent we must have

1 − b + k = x (λ (1 − c + (1 − d) k) + (1 − λ) (1 − b − c + k)) + (1 − x) (1 − b − c + k)

Or, c = xλ (b − dk) (A5)

Observe that we cannot have λ = 0 and so the receiver cannot strictly prefer to choose decline following cry. First, suppose that λ = 1 i.e. that the receiver strictly prefers to choose donate after cry. Hence, x = c/(b − dk) and for the receiver, following an observation of cry, we must have

σH (1 − µ) (1 − d + (1 − c) k) + µ (1 − d + (1 − c) k)

≥ σH (1 − µ) (1 + (1 − b − c) k) + µ (1 + (1 − a − c) k)

using Bayes’ law. This reduces to µ ak − d σ ≤ H 1 − µ d − bk Then,

VR = (1 − µ) ((1 − σH ) (1 + (1 − b) k) + σH x (1 − d + (1 − c) k) + (1 − x) (1 + (1 − b − c) k))

+ µ (x (1 − d + (1 − c) k) + (1 − x) (1 + (1 − a − c) k))

It is easy to see that VR is strictly decreasing in σH . Second, suppose that λ ≤ 1 i.e. that the receiver is indiﬀerent between donate and decline after cry. Accordingly, we must have µ ak − d σ = H 1 − µ d − bk c Since expression A5 must hold, we have b−dk ≤ x ≤ 1 and λ = c/ (x(b − dk)).

ˆ Lemma A.3. Let d ≤ d. There exists an equilibrium (σH , cry; decline, donate) for a non-degenerate

σH if and only if x = c/(b − dk). The receiver’s payoﬀ is strictly decreasing in σH .

Proof. It is clear that R will prefer decline following quiet and donate following cry. θH must be indiﬀerent: hence x (1 − b + k) + (1 − x) (1 + (1 − d) k) = 1 − c + (1 − d) k

which reduces to x = c/(b − dk). It is simple to verify the rest.

Lemma A.4. There exist no equilibria in which θH chooses cry and θN chooses a non-degenerate mixed strategy.

ˆ Proof. First, let d ≥ d. Since θN is mixing, he must be indiﬀerent over his pure strategies in support. Hence, 1 − a − c + k = x (1 + (1 − d) k) + (1 − x) (1 − a + k)

where we have used the fact that following cry the receiver will strictly prefer to choose decline and following quiet the receiver will strictly prefer to choose donate. This reduces to x (dk − a) = c, which is impossible. ˆ Second, let d ≤ d. For θN we must have

1 + (1 − d)k = x (λ (1 − c + (1 − d)k) + (1 − λ) (1 − a − c + k)) + (1 − x) (1 − c + (1 − d)k)

since after quiet the receiver strictly prefers to donate and after cry she may be indiﬀerent (λ ∈ [0, 1]).

This reduces to c = (1 − λ)(dk − a) x, impossible.

ˆ Lemma A.5. Let d ≥ d. There exists an equilibrium (quiet, σN , decline, donate) for a non-degenerate

σN if and only if x = c/(a − dk). The receiver’s payoﬀ is strictly decreasing in σN .

Proof. It is clear that R will prefer decline following quiet and donate following cry. θN must be indiﬀerent: hence x (1 − c + (1 − d) k) + (1 − x) (1 − a − c + k) = 1 − a + k

which reduces to x = c/(a − dk). Again, it is simple to verify the rest.

Lemma A.6. Let d ≤ dˆ.

1. If x < c/(a−dk), there exists no equilibrium in which θN chooses a non-degenerate mixed strategy

and θH chooses quiet.

2. If x = c/(a − dk), there exists a continuum of equilibria (quiet, σN , decline, donate), where σN ≥

1 − (1 − µ)(d − bk)/(µ(ak − d)). The receiver’s payoﬀ is decreasing in σN and her maximal payoﬀ

is VR = 1 − d + k (1 − cµ)+ ck (1 − µ) (d − bk)/(ak − d).

3. If c/(a − dk) ≤ x ≤ 1, then there exists an equilibrium (quiet, σN , decline, donate), where σN = 1 − (1 − µ)(d − bk)/(µ(ak − d)) and λ = 1 − c/ (x(a − dk)). The receiver’s payoﬀ is the same as for 2.

Proof. It is clear that following cry the receiver will strictly prefer to donate. Suppose that the receiver mixes and chooses donate with probability λ following quiet (where λ ∈ [0, 1]), so that this

captures pure strategies for the receiver as well). Then, for θN to be indiﬀerent we must have

1 − c + (1 − d) k = x (λ (1 + (1 − d) k) + (1 − λ) (1 − a + k)) + (1 − x) (1 + (1 − d) k)

Or, c = x (1 − λ)(a − dk) (A6)

Observe that we cannot have λ = 1 and so the receiver cannot strictly prefer to choose donate following quiet. First, suppose that λ = 0 i.e. that the receiver strictly prefers to choose decline after quiet. Hence, x = c/(a − dk) and for the receiver, following an observation of quiet, we must have

(1 − σN ) µ (1 + (1 − a) k) + (1 − µ) (1 + (1 − b) k)

≥ (1 − σN ) µ (1 − d + k) + (1 − µ) (1 − d + k)

using Bayes’ law. This reduces to σN ≥ 1 − (1 − µ)(d − bk)/(µ(ak − d)). Then,

VR = (1 − µ) (x (1 + (1 − b) k) + (1 − x) (1 − d + k))

+ µ [σN (1 − d + (1 − c)k) + (1 − σN )(x (1 + (1 − a) k) + (1 − x) (1 − d + k))]

It is easy to see that VR is strictly decreasing in σN . The maximal value of VR is

VR = (1 − µ) (1 − d + k) + µ (σN (1 − d + (1 − c)k) + (1 − σN ) (1 − d + k))

= 1 − d + k − ckµσN d − bk = 1 − d + k (1 − cµ)+ ck (1 − µ) ak − d Second, suppose that λ ≥ 0 i.e. that the receiver is indiﬀerent between donate and decline after

quiet. Accordingly, we must have σN = 1 − (1 − µ)(d − bk)/(µ(ak − d)). Since expression A6 must hold, we have c/(a − dk) ≤ x ≤ 1 and 1 − λ = c/ (x(a − dk)).

Lemma A.7. There are no equilibria in which both types choose degenerate mixed strategies.

Proof. If d ≥ dˆ (d ≤ dˆ), then following the observation of at least one signal quiet or cry, the receiver must strictly prefer decline (donate). Using this, in conjunction with the fact that both types

of sender must be indiﬀerent over each signal, it is easy to obtain the result.

date: February 24, 2019